Explicit n-descent on elliptic curves, II. Geometry

J. E. Cremona; T. A. Fisher; C. O'Neil; D. Simon; M. Stoll

doi:10.1515/CRELLE.2009.050

Published by De Gruyter June 16, 2009

Explicit n-descent on elliptic curves, II. Geometry

J. E. Cremona , T. A. Fisher , C. O'Neil , D. Simon and M. Stoll

From the journal Journal für die reine und angewandte Mathematik

https://doi.org/10.1515/CRELLE.2009.050

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Abstract

This is the second in a series of papers in which we study the n-Selmer group of an elliptic curve. In this paper, we show how to realize elements of the n-Selmer group explicitly as curves of degree n embedded in ℙ^n–1. The main tool we use is a comparison between an easily obtained embedding into ℙ^n²–1 and another map into ℙ^n²–1 that factors through the Segre embedding ℙ^n–1 × ℙ^n–1 → ℙ^n²–1. The comparison relies on an explicit version of the local-to-global principle for the n-torsion of the Brauer group of the base field.

Received: 2006-12-14

Revised: 2008-02-14

Published Online: 2009-06-16

Published in Print: 2009-July

Explicit n-descent on elliptic curves, II. Geometry

Abstract

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